Pycelle is a dyed-in-the-red Lannister loyalist, and had been the one to stage the quasi-coup that put Kevan in power and began to rebuild the Lannister/Tyrell regime, undoing the damage that Cersei had caused.
Varys wanted him gone because it would further destabilize the government and add to Cersei’s paranoia now that’s she’s the only one left standing.